Inequalities among structure factors

Giacovazzo, C.

doi:10.1107/97809553602060000555

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 217-218 | 1 | 2 |

Section 2.2.5.1. Inequalities among structure factors

C. Giacovazzo^a ^*

^aDipartimento Geomineralogico, Campus Universitario, I-70125 Bari, Italy
Correspondence e-mail: c.giacovazzo@area.ba.cnr.it

2.2.5.1. Inequalities among structure factors

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An extensive system of inequalities exists for the coefficients of a Fourier series which represents a positive function. This can restrict the allowed values for the phases of the s.f.'s in terms of measured structure-factor magnitudes. Harker & Kasper (1948) derived two types of inequalities:

Type 1. A modulus is bound by a combination of structure factors: $[|U_{\bf h}|^{2}\leq {1\over m} \sum\limits_{s = 1}^{m} a_{s} (-{\bf h}) U_{{\bf h} ({\bf I}-{\bf R}_{s})}, \eqno(2.2.5.1)]$ where m is the order of the point group and $[a_{s}(-{\bf h}) =]$ $[\exp (-2\pi i {\bf h}\cdot {\bf T}_{s})]$ .

Applied to low-order space groups, (2.2.5.1) gives $[\eqalign{ P1: &\quad |U_{h, \, k, \, l}|^{2}\leq 1\cr P\bar{1}: &\quad U_{h, \, k, \, l}^{2}\leq 0.5 + 0.5 U_{2h, \, 2k, \, 2l}\cr P2_{1}: &\quad |U_{h, \, k, \, l}|^{2}\leq 0.5 + 0.5(-1)^{k} U_{2h, \, 0, \, 2l}.}]$ The meaning of each inequality is easily understandable: in $[P\bar{1}]$ , for example, $[U_{2h, \, 2k, \, 2l}]$ must be positive if $[|U_{h, \, k, \, l}|]$ is large enough.

Type 2. The modulus of the sum or of the difference of two structure factors is bound by a combination of structure factors: $[ \eqalignno{|U_{\bf h} \pm U_{{\bf h}'}|^{2} &\leq {1\over m} \left\{\sum\limits_{s = 1}^{m} a_{s} (-{\bf h}) U_{{\bf h} ({\bf I}-{\bf R}_{s})} + \sum\limits_{s = 1}^{m} a_{s} (-{\bf h}') U_{{\bf h}' ({\bf I}-{\bf R}_{s})}\right.\cr &\quad \left. \pm 2{\scr Re} \left[\sum\limits_{s = 1}^{m} a_{s} (-{\bf h}') U_{{\bf h}-{\bf h}' {\bf R}_{s}}\right]\right\}&(2.2.5.2)}]$ where $[{\scr Re}]$ stands for `real part of'. Equation (2.2.5.2) applied to P1 gives $[|U_{\bf h}\pm U_{{\bf h}'}|^{2} \leq 2\pm 2 |U_{{\bf h}-{\bf h}'}| \cos \varphi_{{\bf h}-{\bf h}'}.]$

A variant of (2.2.5.2) valid for cs. space groups is $[(U_{\bf h}\pm U_{{\bf h}'})^{2}\leq (1\pm U_{{\bf h}+{\bf h}'}) (1\pm U_{{\bf h}-{\bf h}'}).]$ After Harker & Kasper's contributions, several other inequalities were discovered (Gillis, 1948; Goedkoop, 1950; Okaya & Nitta, 1952; de Wolff & Bouman, 1954; Bouman, 1956; Oda et al., 1961). The most general are the Karle–Hauptman inequalities (Karle & Hauptman, 1950): $[D_{m} = \left|\matrix{U_{0} &U_{-{\bf h}_{1}} &U_{-{\bf h}_{2}} &\ldots &U_{-{\bf h}_{n}}\cr U_{{\bf h}_{1}} &U_{0} &U_{{\bf h}_{1}-{\bf h}_{2}} &\ldots &U_{{\bf h}_{1}-{\bf h}_{n}}\cr U_{{\bf h}_{2}} &U_{{\bf h}_{2}-{\bf h}_{1}} &U_{0} &\ldots &U_{{\bf h}_{2}-{\bf h}_{n}}\cr \vdots &\vdots &\vdots &\ddots &\vdots\cr U_{{\bf h}_{n}} &U_{{\bf h}_{n}-{\bf h}_{1}} &U_{{\bf h}_{n}-{\bf h}_{2}} &\ldots &U_{0}\cr}\right|\geq 0. \eqno(2.2.5.3)]$ The determinant can be of any order but the leading column (or row) must consist of U's with different indices, although, within the column, symmetry-related U's may occur. For [n = 2] and $[{\bf h}_{2} = 2{\bf h}_{1} = 2{\bf h}]$ , equation (2.2.5.3) reduces to $[D_{3} = \left|\matrix{U_{0} &U_{-{\bf h}} &U_{-2{\bf h}}\cr U_{\bf h} &U_{0} &U_{-{\bf h}}\cr U_{2{\bf h}} &U_{\bf h} &U_{0}\cr}\right|\geq 0,]$ which, for cs. structures, gives the Harker & Kasper inequality $[U_{\bf h}^{2} \leq 0.5 + 0.5 U_{2{\bf h}}.]$ For [m = 3] , equation (2.2.5.3) becomes $[D_{3} = \left|\matrix{U_{0} &U_{-{\bf h}} &U_{-{\bf k}}\cr U_{\bf h} &U_{0} &U_{{\bf h}-{\bf k}}\cr U_{\bf k} &U_{{\bf k}-{\bf h}} &U_{0}\cr}\right| \geq 0,]$ from which $[{1 - |U_{\bf h}|^{2} - |U_{\bf k}|^{2} - |U_{{\bf h}-{\bf k}}|^{2} + 2|U_{\bf h} U_{\bf k} U_{{\bf h}-{\bf k}}| \cos \alpha_{{\bf h}, \, {\bf k}} \geq 0,} \eqno(2.2.5.4)]$ where $[\alpha_{{\bf h}, \, {\bf k}} = \varphi_{\bf h} - \varphi_{\bf k} - \varphi_{{\bf h} - {\bf k}}.]$ If the moduli $[|U_{\bf h}|]$ , $[|U_{\bf k}|]$ , $[|U_{{\bf h} - {\bf k}}|]$ are large enough, (2.2.5.4) is not satisfied for all values of $[\alpha_{{\bf h}, \, {\bf k}}]$ . In cs. structures the eventual check that one of the two values of $[\alpha_{{\bf h}, \, {\bf k}}]$ does not satisfy (2.2.5.4) brings about the unambiguous identification of the sign of the product $[U_{\bf h} U_{\bf k} U_{{\bf h} - {\bf k}}]$ .

It was observed (Gillis, 1948) that `there was a number of cases in which both signs satisfied the inequality, one of them by a comfortable margin and the other by only a relatively small margin. In almost all such cases it was the former sign which was the correct one. That suggests that the method may have some power in reserve in the sense that there are still fundamentally stronger inequalities to be discovered'. Today we identify this power in reserve in the use of probability theory.

References

Bouman, J. (1956). A general theory of inequalities. Acta Cryst. 9, 777–780.Google Scholar

Gillis, J. (1948). Structure factor relations and phase determination. Acta Cryst. 1, 76–80.Google Scholar

Goedkoop, J. A. (1950). Remarks on the theory of phase limiting inequalities and equalities. Acta Cryst. 3, 374–378.Google Scholar

Harker, D. & Kasper, J. S. (1948). Phases of Fourier coefficients directly from crystal diffraction data. Acta Cryst. 1, 70–75.Google Scholar

Karle, J. & Hauptman, H. (1950). The phases and magnitudes of the structure factors. Acta Cryst. 3, 181–187.Google Scholar

Oda, T., Naya, S. & Taguchi, I. (1961). Matrix theoretical derivation of inequalities. II. Acta Cryst. 14, 456–458.Google Scholar

Okaya, J. & Nitta, I. (1952). Linear structure factor inequalities and the application to the structure determination of tetragonal ethylenediamine sulphate. Acta Cryst. 5, 564–570.Google Scholar

Wolff, P. M. de & Bouman, J. (1954). A fundamental set of structure factor inequalities. Acta Cryst. 7, 328–333.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 217-218